How to record an object changing its shape?
As you set key frames, be sure to move further down the timeline before setting another key frame. A key frame just sets the value of the properties you are choosing at the current time. If you want to make the changes "live", you can too. In the bottom right of your Timeline, there should be a red button icon that will record your changes and make them key frames as you change it. This isn't quite as advisable IMHO because it gives you less control in the long term, as it marks a new key frame every frame. 

I'd advise using an emphasis on the key part of key frame (har har) by creating key frames with a start value and end value at your start frame and the frame where you want the animation/change to end.

For instance, if you want a sphere to grow in size you would set a frame at the start frame with a set value for scale, then 30 frames later set another key frame with a higher value of its scale. This would look like the sphere slowly (or quickly depending on how far apart the key frames are).

This will work for changing the location of the vertices as well, if you use edit mode and select the specific vertices you want to move and change their location. Mark a start key frame and an end animation key frame where they have moved to your desired position.

Edit based on first comment from OP:

For your use I'd probably just create a particle system if you want a bubbling effect like on a cauldron or soda. If you want to use those objects after the burst out of the main object, you can key frame the objects by right clicking the Location/Rotation or Scale slider in the Object Properties tab, and selecting "Insert Keyframes". You will want to set a keyframe at the start frame, and then keyframe another frame further on where the bubble have come out of the object.